Question

Nathan had an infection, and his doctor wanted him to take penicillin. Because Nathan’s father and paternal grandfather were allergic to penicillin, Nathan has a 75% chance of having the same allergy. The doctor performed a skin test to see whether Nathan would react to it. The test is 98% accurate. What is the probability that Nathan is not allergic to penicillin and the test predicts it?

Answer 1

Answer choices:
0.005

0.015

0.245

0.735

Answer 2

@jim_thompson5910 @SithsAndGiggles

Answer 3

@jim_thompson5910 Can you please find someone?

Answer 4

I found this page

http://en.wikipedia.org/wiki/Accuracy_and_precision

and I'm reading up on it. It might be able to lead us to the answer

Answer 5

@jim_thompson5910 im really confused

Answer 6

now I found this page and I think I'm onto something, so one more second

Answer 7

oops sry

this page

http://vassarstats.net/bayes.html

Answer 8

@jim_thompson5910 do you know what to do?

Answer 9

It seems like something is missing here. Are you able to post a screenshot of this problem?

Answer 10

I got this far, but I'm missing P(R|~A)


A = Nathan is allergic
R = test predicts Nathan has allergies

P(A) = 0.75
P(~A) = 0.25

P(R|A) = 0.98
P(~R|A) = 0.02

P(R) = P(R|A)*P(A) + P(R|~A)*P(~A)
P(R) = 0.98*0.75 + P(R|~A)*0.25

Answer 11

Ok I got this


A = Nathan is allergic
T = test is positive


P(A) = 0.75 (given)
P(~A) = 0.25 (1-0.75)


P(T) = 0.98
P(T|A) = 0.98
P(T|~A) = 0.02



we want to find P(~A|T) which is basically the probability of not having the allergy given the test is positive


P(~A|T) = ( P(T|~A) * P(~A) )/( P(T) )
P(~A|T) = ( 0.02 * 0.25 )/( 0.98 )
P(~A|T) = 0.00510204081632
P(~A|T) = 0.005

Answer 12

so the answer may be 0.005

Answer 13

i just tried that. thats not the answer @jim_thompson5910

Answer 14

then I'm not sure, I'll give it more thought

kropot72 might be able to help though

Answer 15

@kropot72

Answer 16

The probability that Nathan is not allergic to penicillin and the test predicts it is:
\[\large 0.25\times0.02=0.005\]

Answer 17

However you can also interpret the question as asking for the probability that the test correctly predicts that Nathan is not allergic. In this case the probability is:
\[\large 0.25\times0.98=0.245\]

Answer 18

I am sure that my second interpretation is correct: Required probability is 0.245.

Answer 19

Yes you are right. It is 0.245. Thank you!